Vol. 307, No. 2, 2020

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Strong negative type in spheres

Russell Lyons

Vol. 307 (2020), No. 2, 383–390
Abstract

It is known that spheres have negative type, but only subsets with at most one pair of antipodal points have strict negative type. These are conditions on the (angular) distances within any finite subset of points. We show that subsets with at most one pair of antipodal points have strong negative type, a condition on every probability distribution of points. This implies that the function of expected distances to points determines uniquely the probability measure on such a set. It also implies that the distance covariance test for stochastic independence, introduced by Székely, Rizzo and Bakirov, is consistent against all alternatives in such sets. Similarly, it allows tests of goodness of fit, equality of distributions, and hierarchical clustering with angular distances. We prove this by showing an analogue of the Cramér–Wold theorem.

Keywords
Cramér–Wold, hemispheres, expected distances, distance covariance, equality of distributions, goodness of fit, hierarchical clustering.
Mathematical Subject Classification 2010
Primary: 44A12, 45Q05, 51K99, 51M10
Secondary: 62G20, 62H15, 62H20, 62H30
Milestones
Received: 16 May 2019
Revised: 4 April 2020
Accepted: 19 May 2020
Published: 4 September 2020
Authors
Russell Lyons
Department of Mathematics
Indiana Univ, Bloomington
Bloomington, IN
United States